Poker pot odds: you win some, you lose some

If you need a Heart to complete your flush draw the next card can only be one of 4 suits, so your chances are about 25% of getting lucky, right? This isn’t a great approximation but is very easy to calculate. It’s also useful if you like gambling because it overstates your winning probability ..

This article looks at two recent poker hands I played against the same opponent (call him RF), comparing “the rule of 2 and 4" pot odds approximation with the exact pot odds. Were the players getting the required ‘pot odds’ to call?

Relative to the affordable stakes I play (10c/20c NL), they were both ‘big’ pots — 165 big blinds or so. In the first hand, played at the end of August 2022, I won A$33.77 and in the other hand played a few days later I lost $32.33. For reference, this was my only single pot loss of greater than $30 this year in the online game, and one of only 3 wins greater than $30, out of nearly 5000 hands.

Hand #1 (30th of August)

I made a straight over trips, and won $32.23.

Pre-flop

RF has pocket 6s (seated in the HJ),while I have pocket 5s (seated in the CO). Heads up equities are 81%:19% in favour of the 6s. RF raises to $0.66, 4 callers.

Flop ($3.30 pot)

763 rainbow. RF now has trip 6s, while I have a gutshot to a straight.

Exact equities: 85%:15% in favour of the trips.

Approximate equity: without an app in real time to calculate the exact equity, I can quickly note that I have 4 outs to make a straight, and 2 streets to go, 4*2*2 = 16% equity, gives a similar result to 15%. This is the rule of 2 and 4: take the number of outs and multiple by 2 (in percent) to make your draw by the turn, and by 4 to make your draw by the river. This approximation makes sense as there are (about) 50 cards in the deck.

RF now bets $1, I call, the rest fold.

Pot odds: I need 1/5.30 = 19% to make the call, which I don’t have, I only have 15%. Worse, I only have an 8% chance to make my straight by the turn and would probably then need to fold to another bet.

On the other hand, this seems an easy call against a ⅓ of pot bet, given the implied odds. In fact, the expert advice here is to call a small bet, and if I happened to have had an open ended straight draw I should call ‘any reasonably sized bet’, even if it’s quite large. I would also then consider playing aggressively and raising.

Turn ($5.30 pot)

4c, board is now 7643 and I make the straight with my 5s. Lucky …

Exact equities have swung to 25%:75% in favour of the straight now.

Approximate equity: RF has 9 outs for a full house (viz. a 7, 4, or 3); he has 1 out for quads (a 6); finally, he has 2 outs to make the straight (a 5, which we’ll count as 1 out as this only chops the pot). Call it 11 outs, 11 * 2 = 22%, which gives a similar number to the exact 25%.

RF checks, I bet $3.49, RF raises to $10.47, I raise all-in to $30.13, RF calls.

My ⅔ bet is relatively polarising, meaning I have the straight or I probably have a lot less than trips. RF has to call: I could have a higher pocket pair, or 2 pair, or lower trips. Raising my bet seems a bit ambitious though.

Once I raise to $30.13, I almost certainly have a 5.

Pot odds: after my all-in RF now needs to put in another $19.66 to see the river. 19.66/65.56 = 30%. With only 25% equity RF doesn’t have the pot odds to call. Put another way, he has $65.56* 25%= $16.39 equity in the pot, but has to call $19.66, an EV loss of $3.27.

He has no implied odds as I’ve put all my money in, but the other possibility is that I’m bluffing. Is there a 5% chance I’m bluffing that will increase RF’s equity to the required 30%?

Maybe I turned some random nonsense or a single pair into a bluff, or had somehow got to the turn with 2 clubs in my hand, and with the turn being a 2nd club I decided to go in for a big semi-bluff . While this possibility certainly brings RF’s call closer to something reasonable, it’s an under-bluffed spot.

River ($65.56 pot)

9d, a ‘brick’, my straight wins.

Hand #2 (4th of September)

I flopped two pair but my opponent makes a flush on the river.

RF has 97s (clubs) seated in the HJ again; I have AKo in the CO. A 3rd player (initials AM) is also in the hand with A9s in the big blind.

Pre-flop equities: RF (30%), Me (52%) and AM (18%).

Ordinarily A9s is a better hand than 97s, but my AK dominates the A9 hand.

Action limps to me, I raise to $1, 4 callers.

Flop ($5 pot)

Ac, Kc, 6h.

AM flops top pair, I flop 2 pairs, RF flops a flush draw.

Exact equities: AM (<1%, crushed, the run out would need to be the last two 9s in the deck), RF (37%, flush draw), Mike (63%, 2 pair).

Approximate equity: RF has 9 outs to make a flush, as there are 13 club cards in total but we can already see 4 of them. 9 * 4 = 36%, close to the 37%, to make the flush by the river.

AM bets $1.65, call, call, I raise to $4.95, AM and RF call. In this case RF has the pot odds to call (and, as discussed above, should ordinarily call almost any reasonably sized bet with a flush draw).

Turn ($21.50 pot)

2s.

Exact Equities: AM (0% no chance), RF 21%, Mike 79%.

Approximate Equity: RF has 9 outs * 2 = 18%.

AM goes all in $7.67. RF needs pot odds of 7.67/(21.5 + 2* 7.67) = 21%, so he has pot odds (just about) and if I just call so much the better for him (as making a flush will almost always beat both of us).

But I go all-in to $26.38, and RF needs to put in an additional $18.71 to call. He needs pot odds of 18.71/81.93 = 23% here, so with 21% equity RF doesn’t quite have have pot-odds to call. Again, he has no implied odds as both AM and I are all-in, and even if I am bluffing I can probably beat 9 high.

I should comment that having a side-pot complicates the issue, as there are scenarios where RF wins only a portion of the pot.

That said, the pot odds are much closer than expected, and on balance it’s a reasonable call.

River ($81.93 pot)

4c, RF makes the flush and wins a big pot.

Conclusion

Pot odds are far from an exact science. You can’t calculate your equity accurately at the table. The rule of 2 or 4 applied to your number of outs can be calculated in real time, but the result only strictly applies on the turn against an all-in bet. In live play you probably won’t have time to get an exact count of the pot size to calculate your required pot odds. So you’re weighing up between two approximate numbers. Finally, making your flush or draw doesn’t completely guarantee victory.

On the river your actual equity is either 0% or 100%, so the question becomes whether your probability of being ahead is greater than the pot-odds on offer. This depends in turn on whether your opponent over- or under- bluffs, which can’t really be calculated!

Despite these limitations the basic principle is sound however — the worse pot odds you’re being offered the better reason you need to call.